Question

Determine whether the following are subspaces of R2×2 : (a) Thesetofall2×2diagonalmatrices (b) The set of all 2 × 2 triangular matrices (c) The set of all 2 × 2 lower triangular matrices (d) Thesetofall2×2matricesAsuchthat a12=1 (e) The set of all 2 × 2 matrices B such that b11 = 0 (f) The set of all symmetric 2 × 2 matrices (g) Thesetofallsingular2×2matrices

Answer 1

A vector space $V$ is a subspace of a vector space $W$ if

• $V$ is non-empty,
• for any two vectors $v_1,v_2\inV$ we have $v_1+v_2\in V$, and
• for any scalar $k$ and $v\in V$ we have $kv\in V$.

It's easy to show the first condition is met by all the sets in parts (a-g).

(a) is a subspace of $\Bbb R^{2\times2}$ because adding any 2x2 diagonal matrices together, or multiplying one by some scalar, gives another diagonal matrix.

(b) and (c) are also subspaces for the same reasons.

(d) is not a subspace because $\Bbb R^{2\times2}$ because this set of matrices does not contain the zero matrix.

(e), however, is a subspace. Any linear combination of matrices in this set always yields a matrix with 0 in row 1, column 1 entry.

(f) is a subspace. A symmetric matrix is one of the form

$\begin{bmatrix}a&b\\b&c\end{bmatrix}$

Adding two symmetric matrices gives another symmetric matrix:

$\begin{bmatrix}a_1&b_1\\b_1&c_1\end{bmatrix}+\begin{bmatrix}a_2&b_2\\b_2&c_2\end{bmatrix}=\begin{bmatrix}a_1+a_2&b_1+b_2\\b_1+b_2&c_1+c_2\end{bmatrix}$

(g) is not a subspace. Consider the matrices

$\begin{bmatrix}1&0\\0&0\end{bmatrix},\begin{bmatrix}0&0\\0&1\end{bmatrix}$

Both matrices have determinant 0, but their sum is the identity matrix with determinant 1.